A Guide to Hogwarts Arithmancy
by Baelkaz
Summary: Hogwarts third years may schedule this elective. A NEWT is necessary in this subject for Department of Mysteries employment, and an OWL is required for a position in education. Enclosed are key samples of Professor Septima Vector's lesson plans from every year in the curriculum. This is intended as a course supplement, not a replacement. Please support your education professionals.
1. Introduction and Sympathetic Basics

Enclosed herein is a comprehensive lesson plan for the Hogwarts elective, Arithmancy. This elective is considered by the Hogwarts student body to be the most difficult of the options, but is by far the most rewarding. In the same way that learning Latin will assist your spell casting, learning Arithmancy will improve both your practical and theoretical knowledge in near every subject of magical education. This document is intended to provide a basic understanding of Arithmantic concepts. It in no way represents a suitable replacement for enrolling in the Hogwarts Arithmancy course. Professor Vector has submitted her notes from previous years' teaching as a simple review for those who need to brush up on their Sympatheticals, Sequences, or Strings. However, we begin, as is best, at the beginning.

* * *

 **Lesson One:**

Arithmancy is the magic of numbers. As muggles will tell you, numbers can explain everything around us. The same can be said of Arithmancy. All branches of magic find their roots in Arithmancy, and Arithmancy is a required NEWT for application to the Department of Mysteries. Spell creation in particular is a practice that requires an Outstanding to pursue. The first lesson that Hogwarts third years are taught is the primary symbolism of the natural numbers.

 _One_

 _Purity; Birth; Honesty; Light; Rejuvenation; Happiness_

 _Two_

 _Plantlife; Strength; Endurance; Time; Choice; Love_

 _Three_

 _Fungi; Knowledge; Earth; Family; Death; Luck_

 _Four_

 _Animals; Symmetry; Water; Peace; Royalty; Stability_

 _Five_

 _Humanity; Patterns; Ownership; Asymmetry; Air; Wealth_

 _Six_

 _Evil; Temperature; Intent; Poison; Memory; Camaraderie_

 _Seven_

 _Magic; Connection; Aether; Mystery; Fate; Arrogance_

 _Eight_

 _Right; Conception; Astronomy; Childhood; Expression; Faith_

 _Nine_

 _Left; Fire; Ultimatums; Reflection; Confusion; Penance_

One important observation is that each natural number has six common aspects. This is not because of any magical limitation, so much as a human limitation. Six is the number of Intent and Memory. In this way, it is also the number of limitation. Humans have trouble thinking in more than six dimensions. Muggles themselves are usually confined to four. The question that is always asked after this lesson is "What about Zero?" The answer is not so easy. Zero is not a magical number. Zero is the number of nullification. It is the number of anti-magic, and the number of muggles. Some may consider this inequality, looking down on out non-magical counterparts. That is not the truth. Certainly, this Arithmantic theorem has been cited in Pureblood agenda, but it is in no way proof of any superiority. Zero is a hugely important number. In fact, any number multiplied by zero becomes zero. To divide by zero is to create an unsolvable problem. Muggles in this way could in fact be seen as far more powerful than wizards. I leave you today, class, with this list of the natural symbolism to ponder. Your homework is to look for these aspects in your daily life and think about ways they could influence magic as a whole.


	2. Basic Spell Strings

**Lesson Two:**

Welcome back, class. Today, we're going to talk about how the symbolic aspects of natural numbers combine. To make things a little simpler, I'll introduce you to the professional Arithmantic term. The symbolic aspects of natural numbers are called Sympatheticals. To reference the aspects of the number four, for example, is to reference the Fourth Sympathetical. So, we left off last time discussing the purpose of Sympatheticals in daily life, and the possible meanings of Zero. I'm going to address this right now and say that Zero is _not_ called the Zero Sympathetical. Because zero is not a natural number, it has a different name. Zero is what is known as the Null Center, or just the Null. The Center references the inclusion of negative numbers in our theorems, but we won't cover that until your fourth year.

So, to begin with combinations of Sympatheticals, we'll start simple. Some of the combinations may seem obvious, but others are not as much. For example, take two and three. Multiplied, they equal six. This is a good example of how Knowledge combined with Time would equal Memory. Similarly, it is possible that Fungi combined with Choice would equal Poison. However, this combination does not make sense for every possible aspect of a Sympathetical. Plant-life and Luck will very rarely yield Evil. Part of why we study Arithmancy is to be able to understand why certain Sympatheticals work with others. If everything followed conventional wisdom all the time, we could merely hand you a list of the Sympatheticals, pat you on the head, and send you on your way knowing everything about Arithmancy. It does not work like that.

This aspect of Arithmancy is what mostly affects potions and spell creation. Every spell in existence could actually be broken down into a string of numbers and equations. This is what is known as a Spell String, not to be confused with Spell Stringing. Stringing is the practice of smoothly casting spells in a series, flowing from one movement directly into the next in ways that complement each other. Spell Strings are numbers, or, in the case of more complicated spells, full formulas. These tend to depend on factors like Kinesis of the caster, outside conditions, astronomical positioning, and so on. We'll be starting out with basic number strings.

 _Wingardium Leviosa_ is a charm you're all familiar with. It may be no shock to you that the Fifth Sympathetic, representing Air plays a prominent part in this String. The String for _Wingardium Leviosa_ is as follows: 765030. This opens up a few interesting discussion pieces. For example, every spell begins with a Seven. This invokes Magic, as you might suspect. Six almost always follows, save for a handful of exceptions. This invokes Intent. In this case, the content of the unique String begins with Five. As stated, this invokes the Fifth Sympathetical of Air. What follows is an interesting aspect of the Null Center. A Zero in a Spell String begins a series of opposite effects. In this case, the Null effects the Third Sympathetical of Earth, adding to the effect of the Fifth Sympathetical. The second Zero is used as almost a Close to what some liken to parenthesis. A second Zero halts any opposite effect on the enclosed numbers between the Null. This can be used in conjunction with opposing Sympatheticals to increase potency. In this case, combining Air with an absence of Earth will be focused by your Intent to fuel a spell.

In a different example, the _Avada Kedavra_ curse has a fairly simple String as well. It follows as such: 673077104. Analyzed, experts in the DOM have identified that the Sympatheticals at play to be Intent, Magic, Death, Null(Fate, Connection, Rejuvenation), Peace. This curse is a rare example of Intent being given Primacy over Magic. Without the proper feelings of hatred and absolute certainty powering this spell, no magic will respond to the caster's call. The other two Unforgivable Curses are of the same sort, as well as most Dark Magic. What is interesting about this curse is that the Sympathetical for Evil does not appear, as it does in most instances of Dark Magic Strings. It is possible that this is because the _Avada Kedavra_ was used in Arthurian days as a means of peaceful execution.

For the next lesson, we will start on more advanced Strings, breaking into basic formulae.


	3. Applying Arithmancy to Potions

Hello again, students. Today we will be looking at a more practical side to Arithmancy. The study of numbers has long been influential in the magic of potion-making. Discovering the reasons behind steps in brewing will often make you better potioneers.

In truth, potions are some of the most extraordinary magic that wizardkind has discovered. It is a field of combination, taking the disciplines of Astronomy and Arithmancy, and applying them to the magics of Creatures and Herbology. To this day, technomancers are hard at work discovering new ways to apply foreign magics to potions, using Arithmancy to justify their findings.

A simple example of how you might utilize Arithmancy in potion making is found in the action of stirring. You might remember from our first lesson that the Eighth Sympathetical represents "right," while the Ninth Sympathetical represents "left." These also stand for "clockwise" and "anti-clockwise," respectively. Groups of stirring will often be based on this, relying on basic math with the Eighth and Ninth Sympatheticals to infuse the solution with the brewer's magic. In fact, it is the Arithmantic properties of stirring a potion that make it magical at all, as opposed to a poisonous stew.

Let's take for example the basic Cure for Boils potion. I'll transcribe the recipe here for those who have forgotten it.

Finely crush 6 snake fangs into powder

Add 2 measures (half) of snake fang to cauldron

Add 1 finely sliced Punguous Onion

Add scoop of dried nettles

Place mixture on the fire

Add a dash of Flobberworm mucous and stir vigorously anti-clockwise 26 times

The potion should change color to red

Add 1 powdered ginger root and the other half of snake fangs

Stir vigorously clockwise 4 times

Add 3 pickled Shrake spines

Stir gently clockwise 4 times, so as to not agitate the spines

The potion should change color to lavender as the spines dissolve

Drop in 3 horned slugs

Wait 33-44 minutes

Take potion off the fire and add porcupine quills

Stir anti-clockwise 26 times

The potion should settle at a light blue color and have a thin viscosity

There are quite a few things in this simple potion that build off your basic knowledge of Sympatheticals. I'm not asking you to know the properties of the ingredients and how their innate magics interact, but a few obvious things should stick out at you.

All of the ingredients were an odd amount, at either 1 or 3. The snake fangs are initially at 6, but are divided in half so that they become two additions of an odd number. Ideally, counting the individual nettles before adding them and using an odd amount of them would strengthen the potion as well. Can anyone tell me why an odd amount of ingredients is significant? Generally, when dealing with the concepts of "odd" and "even," we represent them with the numbers one and two, respectively. So, knowing that using an odd number of ingredients is invoking the First Sympathetical, does any reasoning come to mind?

The answer is that the First Sympathetical represents rejuvenation, which is important in just about any healing potion. However, due to their complex nature, you will not always find healing potions to use an entirely odd numbered list of ingredients. In fact, part of the reason the Cure for Boils is marked a beginner's level potion is because it does this. The more a potion (or spell) follows common Arithmantic laws (that is, the less exceptions and Null spaces it has,) the easier it is to perform.

The other piece of information I'd like you to take from this potion is the stirring. As we talked about, both clockwise and anti-clockwise are representative of different effects. You may notice that in this potion, you stir anti-clockwise 26 times twice, and clockwise 4 times twice. This makes 54 total stirs anti-clockwise and 8 stirs clockwise. You could theoretically only stir the potion once at 54 rotations, and again with 8 rotations in the other direction. That creates the same basic Arithmantic effect. However, there is a compounding effect from splitting the amount of stirs in half and doing it twice. For every time a potion's stirring is cut in half, its potency is increased, as per the Second Sympathetical of Strength.

To get back on track, there are 54 anti-clockwise stirs. This is 9 times 6, a simple multiplication in this case. Many potions will use operations like the ninth root or something similar to create the correct sympathetic effect. This potion uses 9 times 6, using anti-clockwise stirs to invoke the Sixth Sympathetical of Memory. In this way, the potion "knows" what to restore a person's appearance to, after getting rid of the boils. Furthermore, with 8 clockwise stirs, the math on this one comes out as 1 times 8, using clockwise stirs to invoke the First Sympathetical of Rejuvenation. This compounds with the overall Arithmantic property of odd numbers to increase the healing capabilities.

There are dozens more Arithmantic properties to consider in this potion, including the lack of plant life, relying entirely on creature magic, the natures of the actual creatures themselves, the amount of color changes, the waiting duration and significance of that time. On top of that, there are considerations like what each creature's magic manifests as in those particular pieces of their body, how the Arithmantic strings combine together, the effects of astronomy on this potion, the brewer's intentions and awareness of the task, and so on. In fact, fully analyzing this potion for all of its intricacies (mind you, this is one of the simplest potions on the Hogwarts curriculum) would likely be a full OWL essay question.

Next time, we look at some more advanced combinations of Sympatheticals and what they mean in conjunction, depending on the operations involved.


End file.
